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The Lebesgue Measure
Topic construction of the Lebesgue Measure Seminar 21:00:00 ChanServ changed the topic of #mathematics to: SEMINAR IN PROGRESS. If you want to ask a question, say "!" and wait to be called 20:59:20 ~Mirchiss: somiaj: what do you need to know before hand for the lecture? 21:00:42 somiaj: Mirchiss: it is again going to be mostly definitions and ideas, a bit of real analysis is needed but not much more beyond that. 21:01:58 somiaj: so anyways last time we talked about the abstract structure. For a measure space we have a set Omega, from Omega we have a collection of subests of Omega called Sigma that is a sigma-field (closed under countable unions, intersections and complements) 21:03:04 somiaj: finally we had a function mu: Sigma -> 0,infinity in which was countabally additive (basically meaning the volume of an object is the sum of the volume of its disjoint pieces) 21:03:33 somiaj: so today we are going to talk about creating a measure, in paticularlly the lebesgue measure. 21:04:26 somiaj: The lebesgue measure is created from what is known as an outer measure. If P(Omega) is the power set on omega, then we define mu* : P(Omega) -> 0,infity to be an outer measure if 21:04:44 somiaj: 1) mu*(emptyset) = 0 21:04:59 somiaj: 2) mu*(A) <= mu*(B) whenever A subset B 21:05:25 somiaj: 3) mu* is countably sub-additive, mu*( union_n A_n ) <= sum_n mu*(A_n) 21:06:08 somiaj: so basically an outter measure is a function defeind on the whole powerset that has some of the desired properties. 21:06:23 somiaj: The bigest example of an outer measure if the lebesgue outter measure. 21:07:47 somiaj: To define this outmeausre lets first define an n-cube. an n-cube is a subset of R^n of the form prod_{k=1}^n a_k,b_k 21:08:37 somiaj: where a_k < b_k are real numbers. in R^1 this is just an interval, in R^2 this is a rectangle, in R^3 it is a 'cube' (can't really think of another word but don't think cube means all sides are the same size) 21:09:09 somiaj: and prod_{k=1}^n is the cartiseian product of intervals. 21:10:25 somiaj: We can then define the volume of an n-cube in the standard way. if I = prod_{k=1}^n a_k,b_k then the volume of I is, v(I) = prod_{k=1}^n (b_k - a_k) we are using the usual product on the real numbers) 21:10:43 somiaj: again, in R^1 this gives us length, in R^2 this gives us area, R^3 volume and so forth. 21:11:16 somiaj: From this we can define the outer measure for any A subest R^n. We define this outer measure to be 21:12:21 somiaj: mu*(A) = inf_{A subest union I_k} sum_k v(I_k) where {I_k}_{k a natural number} is any cover of n-cubes of the subest A. 21:13:16 somiaj: (note, I guess to alivate confusion better to call these n-rectangles instead of cubes) 21:15:06 Jafet: ! 21:16:02 somiaj: So we need to show that this creation is an outer measure. First off it is well defined. for any cover of n-rectangles, sum_k v(I_k) is a real number in the range [0,infinity. And if you have a bit of real analysis any non-negative collection of real numbers has an infimum that is also in the range 0,infinity 21:16:07 somiaj: yes Jafet 21:16:14 Jafet: somiaj, that is, mu*(A) is the volume of some arbitrary bunch of n-cubes that happen to contain all the elements in A? 21:16:48 somiaj: Jafet: mu*(A) is the infimum of all possible covers of the subest A. 21:17:22 ichor: ! 21:18:29 somiaj: ichor: yes. 21:19:06 ichor: What is a cover of n-cubes? 21:20:01 somiaj: ichor: {I_k} is a cover of the set A, if A subest union_k I_k 21:20:15 kommodore: I think you want _by_ n-rectangles 21:20:35 somiaj: yea, lets use n-rectangles instead, I think of them as n-cubes, but that is confusing to normal termology. 21:20:35 ichor: somiaj: Ah, I see. Ok. 21:20:48 Jafet: somiaj, "box"? 21:21:03 somiaj: Jafet: that could work as well. 21:21:46 somiaj: So lets say you are given an arbitary set in R^2, to find its area you find the area of countable collection of n-rectangles that cover the set A 21:22:32 somiaj: This is because we can calculate the area of this collection of n-rectangles. We then take the infimum over all possible covers. 21:23:52 somiaj: As I said this forms an outer measure. First off every collection of extended real numbers 0,infinity has an infimum that is also a real number. 21:24:27 somiaj: So we only need to show that it satisfies the three properties I listed for an outermeasure. 21:25:05 somiaj: mu*(emptyset) = 0 is straight forward, consider an empty cover, the volume of the empty cover is 0, thus mu*(emptyset) = 0. 21:25:40 somiaj: second if A subset B, then every cover {I_k} of B is also a cover of A, thus it follows that mu*(A) <= mu*(B) 21:26:39 somiaj: the third is the hardest to show. Suppose {A_n} is a collection of subsets, I need to show that mu*( union_n A_n ) <= sum_n mu*(A_n) 21:27:14 somiaj: this inequality is clear if mu*(A_n) = infinity for any n, so we can assume that mu*(A_n) < infinity for all n. 21:28:09 somiaj: Next we find an 'efficient cover' for each A_n. Give e>0 there exists a cover {I_nk}_{k in N} of A_n such that 21:28:45 somiaj: mu*(A_n) <= sum_k v(I_nk) <= mu*(A_n) + e/2^n 21:29:27 somiaj: i.e. since mu*(A_n) is finite we can find a cover that is within e/2^n of the acual outer measure. 21:30:22 somiaj: Now we notice that {I_nk}_{n in N, k in N} is a countable cover of union_n A_n. 21:31:16 somiaj: thus mu*( union_n A_n ) <= sum_n ( sum_k v(I_nk) ) <= sum_n (mu*(A_n) + e/2^n) = e + sum_n mu*(A_n) 21:31:57 somiaj: since this holds for all e>0, it follows that mu*( union_n A_n ) <= sum_n mu*(A_n) sub-additive 1:34:25 somiaj: yea the indicies n and k just run over the natural numbers. 21:36:40 somiaj: Unfortunately in order to generate an acual measure space we can't just use the outer measure over the whole power set. We have to restrict it. So from the outer measure we restrict to a measure. We say that a subset E is measurable (by the outter measure mu*) if 21:37:13 somiaj: mu*(A) = mu*(A intersect E) + mu*(A - E) for all A subset of Omega 21:37:34 somiaj: This is known as the Caratheodory's Characterization of Measurability. 21:38:29 somiaj: In the case of R^n, not all subests in R^n are measurable, so we have to restrict to ones in which make our outer measure countabally-additive outer measure we only have sub-additivity 21:39:33 somiaj: Equivalently we can say E is measurable if and only if mu*(A union B) = mu*(A) + mu*(B) for all A subest E, and B subset Omega - E 21:42:25 somiaj: It can be proven that the collection of subsets which satisify Caratheodory's Characterization form a sigma-field, and the restriction of mu* to this collection is an countabally additive function. 21:42:41 somiaj: Thus we have a measure space. 21:43:40 somiaj: From here is where the detials begin to get into lots of work. 21:44:25 somiaj: but the idea of the lebesgue measure is we find the 'measure' of a set by covering it with n-rectangles and taking the 'smallest' value of all such covers 21:45:34 somiaj: Though as I mentioned we can do this for any subest of R^n, we cannot create a measure space (since there exist some really bizar subsets of R^n) 21:46:48 somiaj: to prove that this restriction acually forms a sigma-algebra and that mu* under this restriction is countabally additive takes a lot of work and more than we have time for in this siminar. I was just hoping to give the basic idea of how to define the measure 21:47:57 somiaj: Also to prove that there exists non-measurable sets is a theorm from Vitali, which uses zermelo's axiom (variation of the axiom of choice) 21:48:24 kommodore: ! 21:48:28 somiaj: kommodore: yes 21:48:51 kommodore: presumbly the sigma-field given is going to be complete wrt to the measure? 21:49:21 somiaj: correct. 21:49:42 somiaj: One intersting result to get a grasp on what the measurable sets acually look like is to start with the borell sets 21:49:54 somiaj: remember the borell sets are the smallest sigma-field containing the open sets 21:50:13 somiaj: (i.e. all open sets, all closed sets and all countable unions/intersections of such sets) 21:51:05 somiaj: And then to indroduce the concept of a set of measure zero (mu*(A) = 0). Turns out all measure-zero sets and all borell sets are measurable by this critera 21:52:02 somiaj: further more all measurable sets are within a measure-zero set of a borell set 21:52:23 somiaj: though measure zero sets can be quiet bizar, the cantor set is an example of an uncountable set of measure-zero 21:52:59 somiaj: Another useful idea to introduce is what is called G_sigma and F_sigma sets. 21:53:25 somiaj: a G_sigma set is a countable intersection of open sets, and a F_sigma set is a countable union of closed sets. 21:53:52 somiaj: since these sets may neither be open or closed, but must exsist in our borell sigma-field it is nice to give them a name as they become useful in proofs. 21:56:51 somiaj: I don't really have anything else prepared and the hour is up, I can try to answer any questions and hope that for people who havent seen measure theory they now have some idea of how the lebesgue measure is created 21:58:31 _llll_: thanks for the seminar, a good intro 21:59:10 drizzd_: ! 21:59:13 pyninja: thanks 21:59:53 somiaj: drizzd_: yes 22:00:09 drizzd_: did you mean to say: every measureable set is a union of a borel set and a measure-zero set? 22:01:03 somiaj: yea, measure theory is one of those theories that I think is very intuitive at the start, it is a nice generaliziation of volume in R^n, but the detials of the thoery quickly begin to use a lot of analysis tools so lots of places save it for a graduate course 22:01:51 somiaj: drizzd_: the acual theorem for "A characterization of lebesgue measurability" is 22:02:21 somiaj: 1) E subest R^n is lebesuge measurable if and only if E = G - Z, where G is of type G_sigma and Z is a set of measur zero. 22:02:55 somiaj: or 2) E is lebesgue measurable if and only if E = F union Z where F is of type F_sigma and Z is of measure zero 22:03:20 somiaj: again G_sigma is a countable intersection of open sets, and F_sigma is a countable union of closed sets (both borell sets) 22:03:56 lotekk-: somiaj: thank you 22:04:07 drizzd_: I understand 22:04:44 somiaj: so in that light the lebesgue measurable sets are 'almost' borell sets. 22:05:08 zachk: am i correct in assuming there are more bizarre sets of R^n then measurable ones? 22:06:03 somiaj: zachk: yes, it turns out there are some bizar non-measurable sets out there 22:06:07 Jafet: zachk, somiaj mentioned the Vitali set. 22:06:14 somiaj: http://en.wikipedia.org/wiki/Vitali_set is an example of a creation of one for the reals 22:07:00 somiaj: Another example is the sets created in the 22:07:02 somiaj: http://en.wikipedia.org/wiki/Banach-Tarski_paradox 22:07:09 drizzd_: and there is the banach-tarski paradox for R^3 22:07:22 somiaj: all the ones I've seen use the axiom of choice to create them, as in they are not things you can easilly visualize 22:08:06 zachk: so as i go higher up in mathematics it will eventually become improbable for me to visualize alot of constructs? 22:08:11 kommodore: i think you necessarily need AC to construct them 22:09:47 somiaj: zachk: lots of being able to 'visualize' is a personal issue. Some people visualize certain tyes of constructs better than others, for most there are areas in math that you just have to get use to the proofs of the theorms to get a vague idea of the constructs, for others it seems clear. 22:10:42 FunctorSalad: zachk: this is a particularly bad offender though. most things in higher math aren't *that* non-constructive IMHO 22:11:38 kommodore: well, counterexamples are usually bad offenders... 22:12:00 thermoplyae: _llll_: next weekend i'm at a conference and a wedding 22:12:08 FunctorSalad: kommodore: yeah, and this is a counterexample *and* AC-only ;) 22:13:05 drizzd_: I heard that it has been shown in ZF + a large cardinal axiom that it is impossible to prove the existence of lebesgue-measurable sets in ZF. 22:16:38 toed: apparently ZF + notAC has every subset measurable iirc 22:26:46 ChanServ changed the topic of #mathematics to: NEXT SEMINAR: Introductory Riemannian Geometry 1: Differential geometry primer by kommodore on Sunday 3 August 16:00 UTC | Transcript of last seminar: http://www.freenode-math.com/index.php/The_Lebesgue_Measure | Future Seminars: http://www.freenode-math.com/index.php/Seminars Category:Seminar Category:Measure Theory